Whatever the system or the world used, what can a player add to his character background that will help the GM make the game more interesting.

Games are much more interesting with the addition of the character's events/NPCs/whatnot into the main plot(s). Much like Anime, where all the character back stories somewhat interact, it created a sense of wonder. The plot gets richer, the world felt more alive, and mostly the players feel like they are building the story or at least influencing it from the start. Sometimes, I pulled it off, sometimes not so much. So, I am looking for way to make things better for the GM.

I realise that the question is vague and is asking for techniques rather than solid facts but I believe it is a useful question nonetheless -- we shall see how fast it gets closed ^_~

10 Answers
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Have a lot of loose ends. A childhood friend who dissapeared, a mentor figure who turned evil (supposing you are not evil yourself), an unsolved murder in the family, a power your character can't explain, an organisation having a bounty on your character's head...these are just ideas that the GM can play with, which are all bound by the one fact: They have more than one possible endings.

The childhood friend might've changed their name because the local law enforcement organisation was seeking them out, or might've died trying to save someone, or might've left the country to live with their sibling/lover and ended up a royal in their new home. The mentor figure of the past might've become an evil mastermind and powerful villain, or achieved redemption through divine intervention. The power your character can't explain might be in their blood, or in fact, a curse with hidden side effects.

A good idea is to think of a good plot hook for your character, (for example, he is contacted by someone claiming to be their long lost sister, and when your character goes to the place they are supposed to meet, it turns out their sister is an extraplanar being - supposing the setting is fantasy) and then remove the end and leave it up to your GM (so, remove what happens after you go to the meetup place, in our example).

+1 As a GM, the more loose ends, the better. I can even use those to have the party role-play events in a character's past to get everyone engaged.
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F. Randall FarmerDec 9 '11 at 17:15

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I would say that any NPC you have a connection to is a potential loose end anyway, you don't have to go out of your way to attach specific loose ends to them, in fact by doing so you're potentially limiting the kind of plot the GM can use that character for. That said, as other people have commented, such hooks are a good way to indicate to your GM the sort of plot you'd like to be getting, so there's definite advantages to doing both.
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BraibaDec 9 '11 at 18:30

As a player and as a GM, I like handing out/being handed the equivalent of a blank check. In narrative terms, this is an allusion to a past conflict that shows that the player is interested in having that conflict reoccur without the player defining the scope of the conflict or many of the major players.

By leaving the precise details of the backstory vague (blank) it is much easier to involve the character's backstory in the plot of the day. By leaving the magnitude of harm undefined (to a greater or lesser degree) the player is allowing the plot to take front- or back- stage in the day's session.

Fundamentally, the player should exert some narrative control over her player's story, but not restrict details to only her purview. By allowing anyone to add positive or negative elements to the backstory, such that they do not contradict established facts, and by constructing the backstory in such a way to provide narrative hooks for future embellishment, it becomes possible to work a character into plot without too much trouble.

Backgrounds are a great way to help the GM. I once had a character with three ex-wives. The GM exploited that, I assure you. Every so often one of them would make an appearance for some reason or another that would lead to hijinks. Family members are also useful in that manner. Though its important that the GM not overuse it. You dont want the campaign devolve into a weekly mission to rescue a random family member.

Other things might include home towns, past teachers, and involvement in past wars/battles (think of the background in Firefly)

You dont have to be terribly detailed. Just enough to give the GM a nugget of inspiration to run with. It also helps the adventure seem more 'real' if people from your background are coming alive.

The reason many character backgrounds fail to inspire a story is that they are not storytelling backgrounds (backstories), but rather fact sheets. They read like a police file. Subject is 5'6" tall, an elf, from village x, has 3 sisters, and a long-lost uncle. None of these facts speak to the character's, well, character. Even installing a 'life-purpose' doesn't necessarily help. Subject seeks information about long-lost uncle - again, more of a fact.

In narrative, characters have a motivation constructed of three components. First there is a FLAW. Something is wrong with the character who unconsciously seeks to rectify it. What the character seeks is their WANT, the thing they believe will fix their flaw, but is in fact not what they really NEED. When the character gets what he/she NEEDs, the character arch is complete and the story is over.

So if a character seeks a long-lost uncle, the proper backstory answers why the character thinks that finding their uncle will make them happy. Examples:

The character hates (FLAW:hate) his parents (NEED:love) and seeks the parent they always
wanted (WANT), presumably the uncle.

The character hates being an ordinary peasant (FLAW:arrogance), so he seeks a great
purpose (WANT) to prove to the world he's special or a hero (NEED).

The character is haunted (FLAW:regret) by the fact that she sold out her uncle and
seeks redemption (NEED) by saving her uncle (WANT).

Character is defined by choice and motivation. Choice will emerge during game play, but the background contains the motivation. Introducing some motivation can help get past the limits of a fact sheet background.

There have been a lot of good answers above that I have +1'd but I feel that it is important to emphasize that you should try to map out at the very least your immediate family. There have been too many games I have been in where the party is full of orphans because they forget to address that fact/it would be "cool" to be the family avenger. It also can give the players incentive to travel to certain places because they can get a taste of home, or because they know it's safe. It gives the 'blank check' to the DM/GM/ST to evolve those characters as well as let some subplot go to work (captured messages, sibling marrying a rival, etc. ad naseum). Several campaigns I've been in rely on heavy travel, and it's easy to forget where these ties matter.

You should try to add as much as you can "entry-point" for you character.

I mean when you are describing your character and his history you will add some information from childhood, beloved, family, enemies, friends, traveling and so on. Yes, it's nice. But you need to add such episodes which can be entwined easily in your future campaign.

I mean... do not write (i was in %city%. Describe that you did there, purposes of your visit, exit points and so on).

When GM will read your background he should understand the manner of thinking/action. What you like/dislike, what is your personality and so on.

Include links between your character and other important parts of the setting; Other PCs, major NPCs, organizations, etc. Make sure these are ties to other people's stuff. If your character comes with his own Aunt May, that will be boring to the other players. But if his Aunt May is also the evil-Duke-who-we're-fighting's chambermaid, then it's interesting again. Get you character as entangled with the world as you can.

Situations

If you are on the run from the law, and in disguise, then there's always an adventure waiting to happen. Again, try to make sure this is something all the players can participate in, and not just a solo sideshow.